Skip to main content
SpringerLink
Log in

Large deviation analysis of the single server queue

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We establish the large deviation principle (LDP) for the virtual waiting time and queue length processes in the GI/GI/1 queue. The rate functions are found explicitly. As an application, we obtain the logarithmic asymptotics of the probabilities that the virtual waiting time and queue length exceed high levels at large times. Additional new results deal with the LDP for renewal processes and with the derivation of ‘unconditional’ LDPs for ‘conditional ones’. Our approach applies in large deviations ideas and methods of weak convergence theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Abate, G.L. Choudhury and W. Whitt, Waiting-time tail probabilities in queues with long-tail service-time distributions, Queueing Systems 16 (1994) 311–338.

    Google Scholar 

  2. A. de Acosta, Exponential tightness and projective systems in large deviation theory,Volume dedicated to Lucien Le Cam on his 70th birthday.

  3. V. Anantharam, How large delays build up in aGI/G/1 queue, Queueing Systems 5 (1989) 345–367.

    Google Scholar 

  4. S. Asmussen, Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/GI/1 queue, Adv. Appl. Prob. 14 (1982) 143–170.

    Google Scholar 

  5. J.-P. Aubin,L'Analyse non Linéaire et ses Motivations Économiques (Masson, 1984).

  6. J.-P. Aubin and I. Ekeland,Applied Nonlinear Analysis (Wiley, 1984).

  7. P. Billingsley,Convergence of Probability Measures (Wiley, 1968).

  8. A.A. Borovkov, On limit laws for service processes in multi-channel systems (in Russian), Siberian Math. J. 8 (1967) 746–763.

    Google Scholar 

  9. A.A. Borovkov,Stochastic Processes in Queueing Theory (in Russian: Nauka, 1972, English translation: Springer, 1976).

  10. A.A. Borovkov,Asymptotic Methods in Queueing Theory (in Russian, Nauka, 1980).

  11. W. Bryc, Large deviations by the asymptotic value method,Diffusion Processes and Related Problems in Analysis, Vol. 1, ed. M.A. Pinsky (Birkhäuser, 1990) pp. 447–472.

  12. E.G. Coffman, Jr., and M. I. Reiman, Diffusion approximations for computer/communication systems,Math. Comput. Perform. Reliab. (North-Holland, 1984) pp. 33–53.

  13. N.R. Chaganty, Large deviations for joint distributions and statistical applications, Technical Report TR93-2, Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA (1993).

    Google Scholar 

  14. D.A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics 20 (1987) 247–308.

    Google Scholar 

  15. C. Dellacherie,Capacités et Processus Stochastiques (Springer, 1972).

  16. A. Dembo and O. Zeitouni,Large Deviations Techniques and Applications (Jones & Bartlett, 1993).

  17. J.D. Deuschel and D.W. Stroock,Large Deviations (Academic Press, 1989).

  18. I.H. Dinwoodie and S.L. Zabell, Large deviations for exchangeable random vectors, Ann. Prob 20 (1992) 1147–1166.

    Google Scholar 

  19. R.M. Dudley, Convergence of Baire measures, Studia Math. 27 (1966) 251–268.

    Google Scholar 

  20. R.M. Dudley,Real Analysis and Probability (Wadsworth & Brooks/Cole, 1989).

  21. P. Dupuis, H. Ishii and H.M. Soner, A viscosity solution approach to the asymptotic analysis of queueing systems, Ann. Prob. 18 (1990) 226–255.

    Google Scholar 

  22. P. Dupuis and R.S. Ellis, The large deviation principle for a general class of queueing systems, I, Technical Report, Brown University and University of Massachusets (1994).

  23. R.S. Ellis, Large deviations for a general class of random vectors, Ann. Prob. 12 (1984) 1–12.

    Google Scholar 

  24. H. Federer,Geometric Measure Theory (Springer, 1969).

  25. C. Flores, Diffusion approximations for computer communication networks,Comput. Commun., Proc. Symp. Appl. Math., 31 (AMS, Providence, 1985) pp. 83–124.

    Google Scholar 

  26. M.I. Freidlin and A.D. Wentzell,Random Perturbations of Dynamical Systems (in Russian: Nauka, 1979, English translation: Springer, 1984).

  27. J. Gärtner, On large deviations from the invariant measure, Th. Prob Appl. 22 (1977) 24–39.

    Google Scholar 

  28. P.W. Glynn and W. Whitt, Logarithmic asymptotics for steady-state tail probabilities in a single server queue, J. Appl. Prob., Spec. Vol. 31A (1994) 131–156.

    Google Scholar 

  29. P.W. Glynn and W. Whitt, Large deviations behavior of counting processes and their inverses, Queueing Systems 17 (1994) 107–128.

    Google Scholar 

  30. P.W. Glynn, A. Puhalskii and W. Whitt, Functional large deviation principle for inverse processes (in preparation).

  31. P.L. Hennequin et A. Tortrat,Théorie des Probabilités et Quelques Applications (Masson, 1965).

  32. D.L. Iglehart, Limit diffusion approximations for the many server queue and the repairman problem, J. Appl. Prob. 2 (1965) 429–441.

    Google Scholar 

  33. D.L. Iglehart, Extreme values in theGI/GI/1 queue, Ann. Math. Stat. 43 (1972) 627–635.

    Google Scholar 

  34. D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic, I, Adv. Appl. Prob. 2 (1970) 150–177.

    Google Scholar 

  35. D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic, II: sequences, networks and batches, Adv. Appl. Prob. 2 (1970) 355–369.

    Google Scholar 

  36. D.L. Iglehart and W. Whitt, The equivalence of functional central limit theorems for counting processes and associated partial sums, Ann. Math. Stat. 42 (1971) 1372–1378.

    Google Scholar 

  37. N. Ikeda and S. Watanabe,Stochastic Differential Equations and Diffusion Processes (North-Holland, 1981).

  38. J.F.C. Kingman, On queues in heavy traffic, J. Roy. Statist. Soc. B24 (1962) 383–392.

    Google Scholar 

  39. T. Lindvall, Weak convergence of probability measures and random functions in the function spaceD[0, ∞), J. Appl. Prob. 10 (1973) 109–121.

    Google Scholar 

  40. R.Sh. Liptser, Large deviations for a simple closed queueing model, Queueing Systems 14 (1993) 1–31.

    Google Scholar 

  41. R.Sh. Liptser and A. Pukhalskii, Limit theorems on large deviations for semimartingales, Stoch. Stoch. Rep. 38 (1992) 201–249.

    Google Scholar 

  42. J. Lynch and J. Sethuraman, Large deviations for processes with independent increments, Ann. Prob. 15 (1987) 610–627.

    Google Scholar 

  43. P.A. Meyer,Probability and Potentials (Blaisdell, 1966).

  44. A.A. Mogulskii, Large deviations for processes with independent increments, Ann. Prob. 21 (1993) 202–215.

    Google Scholar 

  45. A.A. Mogulskii, Large deviations for trajectories of multidimensional random walks, Th. Prob. Appl. 21 (1976) 300–315.

    Google Scholar 

  46. U. Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971) 518–535.

    Google Scholar 

  47. T. Norberg, Random capacities and their distributions, Prob. Th. Rel. Fields 73 (1986) 281–297.

    Google Scholar 

  48. G.L. O'Brien and W. Vervaat, Capacities, large deviations and loglog laws,Stable Processes, eds. Cambanis, Samorodnitsky and Taqqu (Birkhüser, 1991) pp. 43–83.

  49. G.L. O'Brien, Sequences of capacities, with connections to large deviation theory, Preprint (1994).

  50. G.L. O'Brien and J. Sun, Large deviations on linear spaces, Preprint (1994).

  51. D. Pollard,Convergence of Stochastic Processes (Springer, 1984).

  52. A. Puhalskii, On functional principle of large deviations,New Trends in Probability and Statistics, Vol 1, eds. V. Sazonov and T. Shervashidze (VSP/Moks'las, 1991) pp. 198–218.

  53. A. Puhalskii, On large deviation theory (in Russian), Th. Prob. Appl. 38 (1993) 553–562.

    Google Scholar 

  54. A. Puhalskii, Large deviations of semimartingales via convergence of the predictable characteristics, Stoch. Stoch. Rep. 49 (1994) 27–85.

    Google Scholar 

  55. A. Puhalskii, The method of stochastic exponentials for large deviations, Stoch. Proc. Appl. 54 (1994) 45–70.

    Google Scholar 

  56. A. Puhalskii, Large deviations of the statistical empirical process,Frontiers in Pure and Applied Probability 8, eds. A.N. Shiryaev et al. (TVP, 1995).

  57. A. Puhalskii, Large deviations of semimartingales: a maxingale problem approach (in preparation).

  58. A. Puhalskii, A functional Sanov theorem and large deviations for the infinite server queue,Abstracts, Conf. Appl. Probab. Eng., Comput. Commun. Syst., INRIA/ORSA/TIMS/SMAI, Paris France (1993) p. 136.

    Google Scholar 

  59. Yu.V. Prohorov, Transient phenomena in queueing processes (in Russian), Lit. Mat. Rink. 3 (1963) 199–206.

    Google Scholar 

  60. M.I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res. 9 (1984) 441–458.

    Google Scholar 

  61. R.T. Rockafellar,Convex Analysis (Princeton University Press, 1970).

  62. J.S. Sadowsky and W. Szpankowski, The probability of large queue lengths and waiting times in a heterogeneous multiserver queue, part I: tight limits, Adv. Appl. Prob. 27 (1995) 532–566.

    Google Scholar 

  63. J.S. Sadowsky, The probability of large queue lengths and waiting times in a heterogeneous multiserver queue, part II: positive recurrence and logarithmic limits, Adv. Appl. Prob. 27 (1995) 567–583.

    Google Scholar 

  64. G.E. Shilov and B.L. Gurevich,Integral, Measure and Derivative: a Unified Approach (Prentice Hall, 1966).

  65. A.N. Shiryaev,Probability, 2nd Ed. (in Russian, Nauka, 1989).

  66. A.V. Skorohod, Limit theorems for stochastic processes, Th. Prob. Appl. 1 (1956) 261–292.

    Google Scholar 

  67. C. Stone, Weak convergence of stochastic processes defined on semi-infinite time intervals, Proc. Amer. Math. Soc. 14 (1963) 694–696.

    Google Scholar 

  68. H. Tanaka, Stochastic differential equations with reflecting boundary conditions, Hiroshima Math. J. 9 (1979) 163–177.

    Google Scholar 

  69. S.R.S. Varadhan, Asymptotic probabilities and differential equations, Commun. Pure Appl. Math. 19 (1966) 261–286.

    Google Scholar 

  70. S.R.S. Varadhan,Large Deviations and Applications (SIAM, 1984).

  71. W. Vervaat, Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 23 (1972) 245–253.

    Google Scholar 

  72. W. Vervaat, Narrow and vague convergence of set functions, Stat. Prob. Lett. 6 (1988) 295–298.

    Google Scholar 

  73. A.D. Wentzell,Limit Theorems on Large Deviations for Markov Random Processes (in Russian: Nauka, 1986, English translation: Reidel, 1989).

  74. W. Whitt, Heavy traffic theorems for queues: a survey, Lect. Notes Econ. Math. Syst. 98 (1974) 307–350.

    Google Scholar 

  75. W. Whitt, Some useful functions for functional limit theorems, Math. Oper. Res. 5 (1980) 67–85.

    Google Scholar 

  76. R.L. Dobrushin and E.A. Pechersky, Large deviations for tandem queueing systems, J. Appl. Math. Stoch. Anal. 7 (1994) 301–303.

    Google Scholar 

  77. A. Shwartz and A. Weiss,Large Deviations for Performance Analysis (Chapman & Hall, 1995).

Download references

Author information

Authors and Affiliations

  1. Institute for Problems in Information Transmission, 19, Bolshoi Karetnii, 101447, Moscow, Russia

    A. Puhalskii

Authors
  1. A. Puhalskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported in part by AT&T Bell Labs.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Puhalskii, A. Large deviation analysis of the single server queue. Queueing Syst 21, 5–66 (1995). https://doi.org/10.1007/BF01158574

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01158574

Keywords

Search

Navigation